Algebro-geometric solutions to the lattice potential modified Kadomtsev--Petviashvili equation

TL;DR

This paper constructs algebro-geometric solutions for the lattice potential modified KP equation using Darboux transformations, Lax pairs, and Riemann theta functions, advancing the understanding of integrable discrete systems.

Contribution

It introduces a novel method to derive explicit algebro-geometric solutions for the lpmKP equation via Darboux transformations and Baker-Akhiezer functions.

Findings

Explicit algebro-geometric solutions expressed with Riemann theta functions.

Development of integrable symplectic maps related to the Kaup-Newell spectral problem.

Asymptotic analysis of Baker-Akhiezer functions for the lpmKP equation.

Abstract

Algebro-geometric solutions of the lattice potential modified Kadomtsev-Petviashvili (lpmKP) equation are constructed. A Darboux transformation of the Kaup--Newell spectral problem is employed to generate a Lax triad for the lpmKP equation, as well as to define commutative integrable symplectic maps which generate discrete flows of eigenfunctions. These maps share the same integrals with the finite-dimensional Hamiltonian system associated to the Kaup-Newell spectral problem. We investigate asymptotic behaviors of the Baker-Akhiezer functions and obtain their expression in terms of Riemann theta function. Finally, algebro-geometric solutions for the lpmKP equation are reconstructed from these Baker-Akhiezer functions.